Riemann, Thorin, van Dantzig Pairs, Wald Couples and Hadamard Factorisation
The Hadamard-Weierstrass factorisation of an entire function in the Laguerre-Pólya class is dual to a pair of probabilistic objects: a van Dantzig pair of characteristic functions and a Wald couple of infinitely divisible random variables. The reciprocal of such a function is the Laplace transform of a generalised gamma convolution (GGC) whose Thorin measure encodes the zeros, and Thorin's condition, a real Laplace identity on $(0,\infty)$, is equivalent to the absence of zeros off the critical axis. We prove this duality, together with a closed-form formula for the Thorin density in terms of any Lévy representation. The framework is then applied to the Gamma, hyperbolic, Bessel and Macdonald functions, to the Riemann $ξ$-function, to Dirichlet and modular $L$-functions, and to Dedekind's $η$. For Ramanujan's $τ$ it is set up but not carried through, since reality of the zeros of the associated $L$-function is open. For $ξ$ we prove, unconditionally, that the heat trace $W(t)=\tfrac12\sum_ρ\exp\{(ρ-\tfrac12)^{2}t\}$ is positive and strictly decreasing on $(0,\infty)$, so that $ξ(\tfrac12)/ξ(\tfrac12+\sqrt{u})$ is the Laplace transform of a self-decomposable law, and that $ξ(α)/ξ(α+\sqrt{s})$ is the Laplace transform of a GGC for every $α$ exceeding the real part of every non-trivial zero, hence unconditionally for every $α\ge1$. Complete monotonicity of $W$, equivalently the GGC property at the centre, is shown to be equivalent to the Riemann hypothesis. Self-decomposability is therefore the unconditional ceiling.